Osculating circle

x[t_] := t;
y[t_] := Sin[t]
r[t_] := {x[t], y[t]};

curvature[t_] = Abs[D[D[y[t], t], t]]/(1 + (D[y[t], t])^2)^(3/2);
radius[t_] = If[curvature[t] == 0, 99999, 1/curvature[t]];

T[t_] = D[r[t], t]/Sqrt[D[r[t], t].D[r[t], t]];
Tprime[t_] = D[T[t], t];

circle[t_] := Module[{NNN, center, xc, yc},
  NNN = If[Tprime[t] == {0, 0}, {0, 0},
    Tprime[t]/Sqrt[Tprime[t].Tprime[t]]];

  center = radius[t]*NNN + r[t];
  xc = Extract[center, 1];
  yc = Extract[center, 2];
  {{xc, yc}, radius[t]}
  ];
frame[tt_] :=
 Show[
  Graphics[
   {EdgeForm[Gray], FaceForm[RGBColor[0.95, 0.95, 1.0]],
    Module[{p, r},
     {p, r} = circle[tt];
     {
      Circle[p - {tt, 0}, r],
      Point[p - {tt, 0}],
      Gray,
      Line[{p - {tt, 0}, {0, y[tt]}}]
      }
     ]}
   , PlotRange -> {window {-1, 1}, {-5, 5}}],
  Plot[y[t + tt], {t, -window, window},
   PlotStyle ->
    Directive[Thick, Opacity[0.8], RGBColor[0.2, 0.3, 0.9]]]
  ];
Manipulate[
frame[tt],
{tt,0,2Pi}
]